]>

Γ (z + 1) = \sqrt{2 π} {(z + g + 0.5)}^{z + 0.5} e^{- (z + g + 0.5)} S_{g} (z)

S_{g} (z) = [\frac{1}{2} a_{0} + a_{1} \frac{z}{z + 1} + a_{2} \frac{z (z - 1)}{(z + 1) (z + 2)} + \dots]

a_{k} = {(- 1)}^{k} \sqrt{\frac{2}{π}} e^{g} k \sum_{j = 0}^{k} {(- 1)}^{j} \frac{(k + j - 1)!}{(k - j)! j!} {(\frac{e}{j + g + \frac{1}{2}})}^{j + \frac{1}{2}}

Γ (z) = \frac{{(z + g - 0.5)}^{z - 0.5}}{e^{z + g - 0.5}} L_{g} (z);

L_{g} (z) = C_{0} + \sum_{k = 1}^{N - 1} \frac{C_{N}}{z + k - 1}

D_{i j} = {\begin{matrix} 0 & i f & i \neq j \\ 1 & i f & i = j = 0 \\ - 1 & i f & i = j = 1 \\ \frac{D_{i - 1, i - 1} 2 (2 i - 1)}{i - 1} & o t h e r w i s e \end{matrix}

B_{i, j} = {\begin{matrix} 1 & i f & i = 0 \\ - 1^{j - i} X & i f & i > 0 & j \geq i \\ 0 & o t h e r w i s e \end{matrix}; X = (\begin{matrix} i + j - 1 \\ j - i \end{matrix})

C_{i, j} = {\begin{matrix} \frac{1}{2} & i f & i = j = 0 \\ 0 & i f & j > i \\ - 1^{i - j} S & o t h e r w i s e \end{matrix}; S = \sum_{k = 0}^{i} (\begin{matrix} 2 i \\ 2 k \end{matrix}) (\begin{matrix} k \\ k + j - i \end{matrix})

F_{i} = \frac{(2 i)! e^{i + g + 0.5}}{i! 2^{2 i - 1} {(i + g + 0.5)}^{i + 0.5}}

Γ (z) = {(\frac{z + g - 0.5}{e})}^{z - 0.5} L_{g, e} (z);